This would require it to be 238,900 miles away and travelling roughly
at right angles to the Earth's gravitational field.
Your thinking is correct that an impact would have a very hard time forming the Moon where it is, but the Moon wasn't 238,000 miles away when it formed. The Moon was much closer, about 3-5 Earth radii distant. It's impossible to say precisely how close because there's no way to model it precisely. The interactive tidal forces push the Moon slowly away, and that's still happening today. Working backwards, it's not hard to deduce that 4.4 billion years ago, the Moon was over 10 times closer to the Earth.
Even if the impacting object 'bounced off'
Terminology isn't perfect but a giant impact with incoming velocity of at least 12 or 13 km/s doesn't exactly bounce off. It plows into it, kind of merges into then explodes or rebounds off. Meteor impacts are most accurately represented as explosions and the size of an impact crater tends to be about 10 times the size of the object that impacts. Think about what that means. If the impactor was the size of Mars, or even the size of mercury (estimates vary). A crater 10 times the size of the planet Mercury would be 3 times bigger than the Earth. An impact that large at the very least liquifies the Earth and sends it spinning very fast and perhaps, according to a new theory, doesn't only liquify the Earth but turns it into a kind of temporary gas giant made of vaporized rock called a synestia. Whichever theory you like, it's a lot more involved than just hitting and bounding off. It's a messy splash that turns the Earth into a ball of lava, at the very least.
Another hypothesis is that it took multiple impacts to form the Moon, not a single impact. We don't have any photographs of the event, so all anyone can do is try to find a good model that explains the Moon's formation and the material it's made of. What everyone agrees on, as I said above, is that the Moon was over 10 times closer when it formed. If the Moon is a golfball in the sky to our eyes, then after it formed, it was a volleyball. That would have been quite something to see.
How could a mass of ejecta as large as the Moon all end up travelling
together in a single direction and therefore a single orbit?
Using the synestia model, then what basically happens is the moon forms out of the rotating donut of gas. Collapsing clouds of dust can't form into a single ball in the center because angular momentum needs to be conserved. When the gas that formed our solar-system collapsed there was enough debris with angular momentum to form the planets. Likewise, when Jupiter formed, not all the material fell into Jupiter, where the angular momentum was too high, Jupiter's 4 Galilean moons formed, and, perhaps more than 4 formed initially. Binary stars often form that way. Angular momentum must be conserved, so moons forming out of a rapidly rotating ball/donut of gas is pretty common.
Using the more traditional impact model, which is what you address in your question, it's important to think of it from both frames of reference. There's the surface of the Earth where the impact happens frame of reference and there's the gravitational field or non-rotating frame of reference.
If you're standing on the Earth and you witness a giant meteor impact. Ignoring the fact that you'd die, what you'd see is the surface basically liquify and contract from the impact, then explode outwards. From your frame of reference, material would be blown out in all directions equally, in a simple explosion model. The angle of impact wouldn't change the explosion model from that frame of reference all that much. But what would actually happen is more complicated. The impacting object would have not only straight down velocity but sideways velocity, accelerating the entire crust of the Earth in the direction it impacts. The actual plowing into the Earth would take about 8 minutes before the rebound outwards began, and in those 8 minutes, the Earth's rotation would accelerate significantly. The rotational acceleration would be perhaps 1/2 g force for those 8 minutes and, initially the impact spot and tangential velocity of impact would be shoving it's way across the Earth as the impact continued and the surface of the Earth would rotate faster than the inner part and the axis of rotation would change too, initially just on the surface, and the core would be slower to catch up. It would be an absolutely crazy event to witness, but ignoring all that, it's probably accurate enough to say that the explosion model would shoot debris equally in all directions.
But if you're observing from space, the other frame of reference, lets say, a geostationary orbit (which OK, as the rotation of the planet changes, so does the geostationary distance, but lets say you're just observing from above) for optimal impact viewing, you'd see the Earth's rotation speed increase from the impact. Earth is now rotating about once every 4 hours. Blazingly fast for planetary standards, and as a result, it bulges around the equator and flattens at the poles, visibly flattened, like Jupiter. At the equator relative to the axis of rotation, which, as a result of the impact, the impact spot would be close to the newly positioned equator, in that scenario equator is spinning around at 6 times it's current speed or about 6,000 mph. We usually use km/s for asteroid and escape velocity, so 6,000 mph = about 2.7 km/s.
From your point of view from space, when debris is blown off the Earth from a rotating surface moving at 2.7 km/s, that means, material blown off in the direction of the rotation has a plus 2.7 km/s and material blown off in the opposite direction has a -2.7 km/s. Material blown off perpendicularly, then it would be a vector addition, but the result, material blown roughly off in the direction of rotation is far more likely to make it into orbit and material blown off against the rotation, even if the material is blown away equally in all directions, the material that makes it into orbit would be much more likely to be in line with the Earth's rotation and that's where the tangential velocity comes from.
We usually just call this conservation of angular momentum, but I think, explaining it from the 2 frames of reference can make it easier to visualize. An explosion on a rotating sphere, effects the direction of the ejected orbiting material, where most of the material would orbit with some alignment to the direction to the rotation of the sphere.
So, can I make a moon by simply throwing stuff off the Earth at very high speed?
I wanted to add this point, even if it's not quite what you asked.
Orbits move in ellipses, so, If we ignore air resistance and if you're standing on the surface of the Earth, anything you throw, or, lets say you have a space cannon. Anything you cannon into space, would follow an elliptical orbit and potentially spiral around the Earth and hit you in the back. Of course, the Earth rotates, so that wouldn't really happen, and it wouldn't happen for other reasons as well, like there's not just 2 bodies in space so all orbits are perturbed, and Earth has mascons that would affect the neatness of the orbit.
But, like Newton's cannonball, which in theory, when fired from a high mountain at just the right velocity, would circle the Earth and hit the back of the cannon. Newton focused on the circle or orbit around the Earth, but I always thought that the Cannon shooting itself in the back was a fun aspect too.
An object, shot into orbit with an initial velocity, would either escape, if the escape velocity was high enough (hyperbolic orbit) or it would circle back in an ellipse and return to where it was launched from, because orbits, once set in motion, repeat.
But if you launch heavy enough cannonball into space, lets say, 1/81th the mass of the Earth, then something else happens. You're not just launching a new moon into an orbit that reaches 3-5 Earth radii at it's furthest distance before circling back to the Earth, but you're pushing the Earth away at the same time. Every action has an equal and opposite reaction and 1/81th the Mass of the Earth is enough mass where you have to consider the push on both objects, not just the theoretical cannonball.
In this scenario, when your new moon spiraled back to where you launched it, the Earth would have moved about 150 miles and it wouldn't be where it was. Earth would be lighter than it was when the object was launched. The two objects might still collide as each would in effect orbit the other, but you'd no longer have a neat Kepler orbit, but something a little more mathematically complicated when both masses need to be considered. (That's why I said 12 or 13 km/s above, not Earth's escape velocity, because both Earth and Theia would have escape velocities, so the minimum impact speed is slightly higher than for a standard space rock).
Kepler's orbits and Newton's calculated orbits assume the mass ratio is 100 to 0. The math changes when you consider that both objects have mass. In the case of the giant impact, enough mass was blown off the Earth that the Kepler eclipse model doesn't quite apply. But that's not the primary factor that made it possible for the Moon to form with a close to circular orbit.
The newly formed debris shot into orbit has enough mass to have a gravitational effect on itself, and that you have collisions, so, individual bits of debris entered orbit with elliptical orbits the combination of debris, after collisions and gravitational interaction doesn't retain high eccentricity orbits. A lot of the ejecta, perhaps most of it, probably did fall back to the Earth, but enough entered a temporary ring-like system of orbiting cloud of debris that the Moon was able to form, and it likely formed with a reasonably circular orbit too, because the debris, by collision and gravitational interaction, corrected for a lot of individual bits' eccentric orbits.
Similar to how the rings of Saturn which are neat and circular. Non circular orbits in a crowded debris field would be more likely to collide, so over time, individual bits of debris that remained in orbit would move towards more circular orbits in a similar way that enough material tends to spiral into a disk, because variations in inclination or eccentricity would tend to cancel out over time.
I'd be remiss if I didn't point out that the giant impact theory has some issues, it was still a brilliant theory and it would be more accurate to say that it needed some adjustments, so, I don't like saying it was wrong. I think it was brilliant, but there are problems working out the details with the debris field formation of the Moon, or so I've read, but the Moon having a circular orbit from formation isn't one of those problems. That part of the model works.
I can't show you the numbers because you need a super-computer to run those numbers and that's above my pay-grade, so I tried to explain it in a way that can be visualized. Hope that helps.
